Self-similar collapse of a self-gravitating viscous disk

A self-similar solution for time evolution of isothermal, self-gravitating viscous disks is found under the condition that alpha'=alpha(H/r) is constant in space (where a is the viscosity parameter and H/r is the ratio of a half-thickness to the radius of the disk). This solution describes a homologous collapse of a disk via self-gravity and viscosity. The disk structure and evolution are distinct in the inner and outer parts. There is a constant mass inflow in the outer portions so that the disk has flat rotation velocity, constant accretion velocity, and surface density decreasing outward according to Sigma proportional to r(-1). In the inner portions, in contrast, mass is accumulated near the center owing to the boundary condition of no radial velocity at the origin, a strong central concentration being thereby produced; surface density varies according to Sigma proportional to(-5/3). Moreover, the transition radius separating the inner and outer portions increases linearly with time. The consequence of such a high condensation is briefly discussed in the context of formation of a quasar black hole.